A remarkable contribution of the above result lies in the fact that \(M_{\Phi _p},M_{\Psi _q}\)  are explicitly determined to be of Berger, Coburn and Lebow type isometries [1]}. In particular, there exists a collection \((\mathcal {E},U,P)\)  where, \(\mathcal {E}\)  is a Hilbert space, \(U\)  is a unitary and \(P\)  is an orthogonal projection such that
\(\Phi _p(\mathbf {z}) = (P+z_pP^{\perp })U^*; \quad \Phi _q(\mathbf {z}) = U(P^{\perp } + z_p P).\) 
